= Hecke operators are self-adjoint for the Petersson inner product
{c}
{title2=$\langle T_nf,g\rangle=\langle f,T_ng\rangle$}
On level-one <cusp forms>, determinant-normalized <Hecke operators> satisfy $\langle T_nf,g\rangle=\langle f,T_ng\rangle$. Unfolding their finite correspondences transfers a matrix to its inverse; its adjugate has the same determinant and reverses the correspondence. Since these operators commute, the cusp space has a common <orthonormal basis> of <Hecke eigenforms>, with real eigenvalues.
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